A Hands-On Approach to Operations and Equivalent Expressions

Henri Picciotto

About this session:

The Lab Gear manipulatives help students transition from numbers to variables, and from the concrete to the abstract, through activities that promote an understanding of operations: the distributive law, combining like terms, factoring, etc. This is a necessary prerequisite to understanding equivalent expressions, a key concept in middle school.

Participants will work through activities which develop a visual / geometric model of the basic structures of algebra. This enhances classroom discourse: it provides an entry point for students who find it difficult to generalize the ideas of arithmetic, and at the same time deepens and enriches the understanding of the more talented. The inner logic of the model replaces the memorization of seemingly arbitrary rules, students work cooperatively, and they learn to communicate about math.

First, some puzzles: make rectangles with the Lab Gear blocks; measure the dimensions in terms of the variables; write “length times width = area” equations. Discussion about this (and 3D volume puzzles) leads to the distributive rule and factoring. Next: find the perimeters of Lab Gear figures, which leads to the need to combine like terms to facilitate communication. Finally: solve “make a square” puzzles, laying the groundwork for completing the square and the quadratic formula.

Participants will respond to prompts and challenges by using the blocks, with the option to discuss their solutions with their neighbors. I will periodically interrupt them to summarize key concepts and suggest ways to make the transition from hands-on to pencil and paper, stressing that manipulatives are a means, not an end. I will also discuss the practicalities of manipulatives in the classroom, and how those activities might be sequenced in relation to the rest of the curriculum.

I’m happy to respond to comments.

More information about the Lab Gear.

More information about me.


Henri Picciotto

Henri Picciotto

Henri Picciotto, a math education consultant and author, has retired from the classroom after 42 years of teaching at every level from counting to calculus. He is interested in puzzles, manipulatives, and technological tools, and shares his ideas on www.MathEducationPage.org.

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